2.3 Specific heuristics

This list was written by A.H. Schoenfeld:

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  draw a diagram (if at all possible)

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  examine special cases

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    choose special values to exemplify the problem and get a "feel" for it

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    examine limiting cases to explore the range of possibilities

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    set any integer parameters equal to $1,2,3,...$ in sequences and look for an inductive pattern.

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  try to simplify the problem

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    exploiting symmetry

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    without loss of generality

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  consider essentially equivalent problems

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  replace the conditions with equivalent ones

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  recombine the elements of the problem in different ways

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  introduce auxiliary elements (e.g. substitutions)

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  reformulate the problem

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  change of perspective or notation

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  considering argument by contradiction or contrapositive

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  assuming you have a solution, and determining its properties (i.e. what would a valid answer look like)

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  consider slightly modified problems

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    choose sub-goals (obtain partial fulfilment of the conditions)

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    relax a condition and then try to re-impose it

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    decompose the domain of the problem and work on it case by case

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  consider broadly modified problems

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    construct an analogous problem with fewer variables

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    hold all but one variable fixed to determine that variable’s impact

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    try to exploit any related problems which have similar

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      form

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      "givens"

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      conclusions

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  verifying solutions

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    Does it pass these simple tests?

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      Does it use all the pertinent data?

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      Does it conform to reasonable estimates or predictions?

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      Does it withstand tests of symmetry, dimension analysis, or scaling?

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    Does it pass these general tests?

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      Can it be obtained differently?

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      Can it be substantiated by special cases?

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      Can it be reduced to known results?

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      Can it be used to generate something you know?